函数名 功能
Language 求已知数据点的拉格朗日插值多项式
Atken 求已知数据点的艾特肯插值多项式
Newton 求已知数据点的均差形式的牛顿插值多项式
Newtonforward 求已知数据点的前向牛顿差分插值多项式
Newtonback 求已知数据点的后向牛顿差分插值多项式
Gauss 求已知数据点的高斯插值多项式
Hermite 求已知数据点的埃尔米特插值多项式
SubHermite 求已知数据点的分段三次埃尔米特插值多项式及其插值点处的值
SecSample 求已知数据点的二次样条插值多项式及其插值点处的值
ThrSample1 求已知数据点的第一类三次样条插值多项式及其插值点处的值
ThrSample2 求已知数据点的第二类三次样条插值多项式及其插值点处的值
ThrSample3 求已知数据点的第三类三次样条插值多项式及其插值点处的值
BSample 求已知数据点的第一类B样条的插值
DCS 用倒差商算法求已知数据点的有理分式形式的插值分式
Neville 用Neville算法求已知数据点的有理分式形式的插值分式
FCZ 用倒差商算法求已知数据点的有理分式形式的插值分式
DL 用双线性插值求已知点的插值
DTL 用二元三点拉格朗日插值求已知点的插值
DH 用分片双三次埃尔米特插值求插值点的z坐标
function f = Atken(x,y,x0)
syms t;
if(length(x) == length(y))
n = length(x);
else
disp('x和y的维数不相等！');
return;
end %检错
y1(1:n) = t; %符号函数数组要赋初值
for(i=1:n-1)
for(j=i+1:n)
y1(j) = y(j)*(t-x(i))/(x(j)-x(i))+y(i)*(t-x(j))/(x(i)-x(j));
end
y = y1;
simplify(y1);
end
if(nargin == 3)
f = subs(y1(n),'t',x0); %计算插值点的函数值
else
simplify(y1(n)); %化简
f = collect(y1(n)); %将插值多项式展开
f = vpa(f,6); %将插值多项式的系数化成6位精度的小数
end
function f0 = BSample(a,b,n,y,y_1,y_N,x0)
f0 = 0.0;
h = (b-a)/n;
c = zeros(n+3,1);
b = zeros(n+1,1);
for i=0:n-1
if(a+i*h<=x0) && (a+i*h+h>=x0)
index = i;
break;
end
end %找到x0所在区间
A = diag(4*ones(n+1,1));
I = eye(n+1,n+1);
AL = [I(2:n+1,:);zeros(1,n+1)];
AU = [zeros(1,n+1);I(1:n,:)];
A = A+AL+AU; %形成系数矩阵
for i=2:n
b(i,1) = 6*y(i);
end
b(1) = 6*y(1)+2*h*y_1;
b(n+1) = 6*y(n+1)-2*h*y_N;
d = followup(A,b); %用追赶法求出系数
c(2:n+2) = d;
c(1) = c(2) - 2*h*y_1; %c(-1)
c(n+3) = c(3)+2*h*y_N; %c(n+1)
x1 = (a+index*h-h-x0)/h;
m1 = c(index+1)*(-((abs(x1))^3)/6+(x1)^2-2*abs(x1)+4/3);
x2 = (a+index*h-x0)/h;
m2 = c(index+2)*((abs(x2))^3/2-(x2)^2+2/3);
x3 = (a+index*h+h-x0)/h;
m3 = c(index+3)*((abs(x3))^3/2-(x3)^2+2/3);
x4 = (a+index*h+2*h-x0)/h;
m4 = c(index+4)*(-((abs(x4))^3)/6+(x4)^2-2*abs(x4)+4/3);
f0 = m1+m2+m3+m4; %求出插值
function f = DCS(x,y,x0)
syms t;
if(length(x) == length(y))
n = length(x);
c(1:n) = 0.0;
else
disp('x和y的维数不相等！');
return;
end
c(1) = y(1);
for(i=1:n-1)
for(j=i+1:n)
y1(j) = (x(j)-x(i))/(y(j)-y(i));
end
c(i+1) = y1(i+1);
y = y1;
end
f = c(n);
for(i=1:n-1)
f = c(n-i) + (t-x(n-i))/f;
f = vpa(f,6);
if(i==n-1)
if(nargin == 3)
f = subs(f,'t',x0);
else
f = vpa(f,6);
end
end
end;
function fz = DH(x,y,x0,y0,zx,zy,zxy)
n = length(x);
m = length(y);
for i=1:n
if(x(i)<=x0)&& (x(i+1)>=x0)
index_x = i;
break;
end
end %找到x0所在区间
for i=1:m
if(y(i)<=y0)&& (y(i+1)>=y0)
index_y = i;
break;
end
end %找到y0所在区间
hx = x(index_x+1) - x(index_x);
hy = y(index_y+1) - y(index_y);
tx = (x0 - x(index_x))/hx; %插值坐标归一化
ty = (y0 - y(index_y))/hy; %插值坐标归一化
Hl = [(1-tx)^2*(1+2*tx) tx*tx*(3-2*tx) tx*(1-tx)^2 tx*tx*(tx-1)]; %左向量
Hr = [(1-ty)^2*(1+2*ty); ty*ty*(3-2*ty); ty*(1-ty)^2 ; ty*ty*(ty-1)]; %右向量
C = [Z(index_x, index_y) Z(index_x,index_y+1) zy(index_x, index_y) ...
zy(index_x, index_y+1);
Z(index_x+1, index_y) Z(index_x+1,index_y+1) zy(index_x+1, index_y) ...
zy(index_x+1, index_y+1);
zx(index_x, index_y) zy(index_x, index_y+1) zxy(index_x, index_y) ...
zxy(index_x, index_y+1);
zx(index_x+1, index_y) zy(index_x+1, index_y+1) zxy(index_x+1, index_y) ...
zxy(index_x+1, index_y+1)]; %C矩阵
fz = Hl*C*Hr;
function fz = DL(x,y,Z,x0,y0,eps)
n = length(x);
m = length(y);
for i=1:n
if(x(i)<=x0)&& (x(i+1)>=x0)
index_x = i;
break;
end
end %找到x0所在区间
for i=1:m
if(y(i)<=y0)&& (y(i+1)>=y0)
index_y = i;
break;
end
end %找到y0所在区间
A = [1 x(index_x) y(index_y) x(index_x)* y(index_y);
1 x(index_x+1) y(index_y+1) x(index_x+1)* y(index_y+1);
1 x(index_x) y(index_y+1) x(index_x)* y(index_y+1);
1 x(index_x+1) y(index_y) x(index_x+1)* y(index_y)];
iA = inv(A);
B = iA*[Z(index_x,index_y); Z(index_x+1,index_y+1); Z(index_x,index_y+1);
Z(index_x+1,index_y)];
fz = [1 x0 y0 x0*y0]*B;
function fz = DTL(x,y,Z,x0,y0)
syms s t;
f = 0.0;
n = length(x);
m = length(y);
for i=1:n
if(x(i)<=x0)&& (x(i+1)>=x0)
index_x = i;
break;
end
end %找到x0所在区间
for i=1:m
if(y(i)<=y0)&& (y(i+1)>=y0)
index_y = i;
break;
end
end %找到y0所在区间
if index_x == 1
cx(1:3) = index_x:(index_x+2);
else
if index_x == n-1
cx(1:3) = (index_x-1):(index_x+1);
else
if abs(x(index_x-1)-x0)>abs(x(index_x+2)-x0)
cx(1:3) = (index_x):(index_x+2);
else
cx(1:3) = (index_x-1):(index_x+1);
end
end
end %找到离x0最近的三个x坐标
if index_y == 1
cy(1:3) = index_y:(index_y+2);
else
if index_y == m-1
cy(1:3) = (index_y-1):(index_y+1);
else
if abs(y(index_y-1)-y0)>=abs(y(index_y+2)-y0)
cy(1:3) = (index_y):(index_y+2);
else
cy(1:3) = (index_y-1):(index_y+1);
end
end
end %找到离y0最近的三个y坐标
for i=1:3
i1 = mod(i+1,3);
if(i1 == 0)
i1 = 3;
end
i2 = mod(i+2,3);
if(i2 == 0)
i2 = 3;
end
for j=1:3
j1 = mod(j+1,3);
if(j1 == 0)
j1 = 3;
end
j2 = mod(j+2,3);
if(j2 == 0)
j2 = 3;
end
f = f+Z(cx(i),cy(j))*((t-x(cx(i1)))*(t-x(cx(i2)))/(x(cx(i))-x(cx(i1)))/(x(cx(i))-x(cx(i2))))* ...
(s-y(cy(j1)))*(s-y(cy(j2)))/(y(cy(j))-y(cy(j1)))/(y(cy(j))-y(cy(j2)));
%插值多项式
end
end
fz = subs(f,'[t s]',[x0 y0]);
function [f,x0] = FCZ(x,y,y0)
syms t;
if(length(x) == length(y))
n = length(x);
c(1:n) = 0.0;
else
disp('x和y的维数不相等！');
return;
end
c(1) = x(1);
y1 = x;
for(i=1:n-1)
for(j=i+1:n)
y2(j) = (y1(j)-y1(i))/(y(j)-y(i));
end
c(i+1) = y2(i+1);
y1 = y2;
end
f = c(1);
for(i=1:n-1)
ff = c(i+1);
for(j=1:i)
ff = ff*(t-y(j));
end
f = f + ff;
end;
x0 = subs(f,'t',y0);
function f = Gauss(x,y,x0)
if(length(x) == length(y))
n = length(x);
else
disp('x和y的维数不相等！');
return;
end
xx =linspace(x(1),x(n),(x(2)-x(1)));
if(xx ~= x)
disp('节点之间不是等距的！');
return;
end
if( mod(n,2) ==1)
if(nargin == 2)
f = GStirling(x,y,n);
else if(nargin == 3)
f = GStirling(x,y,n,x0);
end
end
else
if(nargin == 2)
f = GBessel(x,y,n);
else if(nargin == 3)
f = GBessel(x,y,n,x0);
end
end
end
function f = GStirling(x,y,n,x0)
syms t;
nn = (n+1)/2;
f = y(nn);
for(i=1:n-1)
for(j=i+1:n)
y1(j) = y(j)-y(j-1);
end
if(mod(i,2)==1)
c(i) = (y1((i+n)/2)+y1((i+n+2)/2))/2;
else
c(i) = y1((i+n+1)/2)/2;
end
if(mod(i,2)==1)
l = t+(i-1)/2;
for(k=1:i-1)
l = l*(t+(i-1)/2-k);
end
else
l_1 = t+i/2-1;
l_2 = t+i/2;
for(k=1:i-1)
l_1 = l_1*(t+i/2-1-k);
l_2 = l_2*(t+i/2-k);
end
l = l_1 + l_2;
end
l = l/factorial(i);
f = f + c(i)*l;
simplify(f);
f = vpa(f, 6);
y = y1;
if(i==n-1)
if(nargin == 4)
f = subs(f,'t',(x0-x(nn))/(x(2)-x(1)));
end
end
end
function f = GBessel(x,y,n,x0)
syms t;
nn = n/2;
f = (y(nn)+y(nn+1))/2;
for(i=1:n-1)
for(j=i+1:n)
y1(j) = y(j)-y(j-1);
end
if(mod(i,2)==1)
c(i) = y1((i+n+1)/2)/2;
else
c(i) = (y1((i+n)/2)+y1((i+n+2)/2))/2;
end
if(mod(i,2)==0)
l = t+i/2-1;
for(k=1:i-1)
l = l*(t+i/2-1-k);
end
else
l_1 = t+(i-1)/2;
l_2 = t+(i-1)/2-1;
for(k=1:i-1)
l_1 = l_1*(t+(i-1)/2-k);
l_2 = l_2*(t+(i-1)/2-1-k);
end
l = l_1 + l_2;
end
l = l/factorial(i);
f = f + c(i)*l;
simplify(f);
f = vpa(f, 6);
y = y1;
if(i==n-1)
if(nargin == 4)
f = subs(f,'t',(x0-x(nn))/(x(2)-x(1)));
end
end
end
function f = Hermite(x,y,y_1,x0)
syms t;
f = 0.0;
if(length(x) == length(y))
if(length(y) == length(y_1))
n = length(x);
else
disp('y和y的导数的维数不相等！');
return;
end
else
disp('x和y的维数不相等！');
return;
end
for i=1:n
h = 1.0;
a = 0.0;
for j=1:n
if( j ~= i)
h = h*(t-x(j))^2/((x(i)-x(j))^2);
a = a + 1/(x(i)-x(j));
end
end
f = f + h*((x(i)-t)*(2*a*y(i)-y_1(i))+y(i));
if(i==n)
if(nargin == 4)
f = subs(f,'t',x0);
else
f = vpa(f,6);
end
end
end
function f = Language(x,y,x0)
syms t;
if(length(x) == length(y))
n = length(x);
else
disp('x和y的维数不相等！');
return;
end %检错
f = 0.0;
for(i = 1:n)
l = y(i);
for(j = 1:i-1)
l = l*(t-x(j))/(x(i)-x(j));
end;
for(j = i+1:n)
l = l*(t-x(j))/(x(i)-x(j)); %计算拉格朗日基函数
end;
f = f + l; %计算拉格朗日插值函数
simplify(f); %化简
if(i==n)
if(nargin == 3)
f = subs(f,'t',x0); %计算插值点的函数值
else
f = collect(f); %将插值多项式展开
f = vpa(f,6); %将插值多项式的系数化成6位精度的小数
end
end
end
function f = Neville(x,y,x0)
syms t;
if(length(x) == length(y))
n = length(x);
else
disp('x和y的维数不相等！');
return;
end
y1(1:n) = t;
for(i=1:n-1)
for(j=i+1:n)
if(j==2)
y1(j) = y(j)+(y(j)-y(j-1))/((t-x(j-i))/(t-x(j)))*(1-(y(j)-y(j-1))/y(j));
else
y1(j) = y(j)+(y(j)-y(j-1))/((t-x(j-i))/(t-x(j)))*(1-(y(j)-y(j-1))/(y(j)-y(j-2)));
end
end
y = y1;
if(i==n-1)
if(nargin == 3)
f = subs(y(n-1),'t',x0);
else
f = vpa(y(n-1),6);
end
end
end
function f = Newton(x,y,x0)
syms t;
if(length(x) == length(y))
n = length(x);
c(1:n) = 0.0;
else
disp('x和y的维数不相等！');
return;
end
f = y(1);
y1 = 0;
l = 1;
for(i=1:n-1)
for(j=i+1:n)
y1(j) = (y(j)-y(i))/(x(j)-x(i));
end
c(i) = y1(i+1);
l = l*(t-x(i));
f = f + c(i)*l;
simplify(f);
y = y1;
if(i==n-1)
if(nargin == 3)
f = subs(f,'t',x0);
else
f = collect(f); %将插值多项式展开
f = vpa(f, 6);
end
end
end
function f = Newtonback(x,y,x0)
syms t;
if(length(x) == length(y))
n = length(x);
c(1:n) = 0.0;
else
disp('x和y的维数不相等！');
return;
end
f = y(n);
y1 = 0;
xx =linspace(x(1),x(n),(x(2)-x(1)));
if(xx ~= x)
disp('节点之间不是等距的！');
return;
end
for(i=1:n-1)
for(j=i+1:n)
y1(j) = y(j)-y(j-1);
end
c(i) = y1(n);
l = t;
for(k=1:i-1)
l = l*(t+k);
end;
f = f + c(i)*l/factorial(i);
simplify(f);
y = y1;
if(i==n-1)
if(nargin == 3)
f = subs(f,'t',(x0-x(n))/(x(2)-x(1)));
else
f = collect(f);
f = vpa(f, 6);
end
end
end
function f = Newtonforward(x,y,x0)
syms t;
if(length(x) == length(y))
n = length(x);
c(1:n) = 0.0;
else
disp('x和y的维数不相等！');
return;
end
f = y(1);
y1 = 0;
xx =linspace(x(1),x(n),(x(2)-x(1)));
if(xx ~= x)
disp('节点之间不是等距的！');
return;
end
for(i=1:n-1)
for(j=1:n-i)
y1(j) = y(j+1)-y(j);
end
c(i) = y1(1);
l = t;
for(k=1:i-1)
l = l*(t-k);
end;
f = f + c(i)*l/factorial(i);
simplify(f);
y = y1;
if(i==n-1)
if(nargin == 3)
f = subs(f,'t',(x0-x(1))/(x(2)-x(1)));
else
f = collect(f);
f = vpa(f, 6);
end
end
end
function [f,f0,fd0] = SecSample (x,y,y_1,x0)
syms t;
f = 0.0;
f0 = 0.0;
if(length(x) == length(y))
if(length(y) == length(y_1))
n = length(x);
else
disp('y和y的导数的维数不相等！');
return;
end
else
disp('x和y的维数不相等！');
return;
end %维数检查
for i=1:n
if(x(i)<=x0)&& (x(i+1)>=x0)
index = i;
break;
end
end %找到x0所在区间
d = y_1(1)*(x(2)-x(1))/2+y(1);
for i=2:n-1
d = d + y_1(i)*(x(i+1)-x(i-1))/2;
end
h = x(index+1) - x(index); %x0所在区间长度
f = y_1(index+1)*(t-x(index))^2/2/h + ...
y_1(index)*(t-x(index+1))^2/2/h + d; %x0所在区间的插值函数
fd = (t-x(index))*y_1(index+1)/h + y_1(index)*(x(index+1)-t)/h;
%x0所在区间的插值函数的导数
f0 = subs(f,'t',x0); %x0处的插值
fd0 = subs(fd,'t',x0); % x0处的导数插值
function [f,f0] = SubHermite(x,y,y_1,x0)
syms t;
f = 0.0;
f0 = 0.0;
if(length(x) == length(y))
if(length(y) == length(y_1))
n = length(x);
else
disp('y和y的导数的维数不相等！');
return;
end
else
disp('x和y的维数不相等！');
return;
end %维数检查
for i=1:n
if(x(i)<=x0)&& (x(i+1)>=x0)
index = i;
break;
end
end %找到x0所在区间
h = x(index+1) - x(index); %x0所在区间长度
fl = y(index)*(1+2*(t-x(index))/h)*(t-x(index+1))^2/h/h + ...
y(index+1)*(1-2*(t-x(index+1))/h)*(t-x(index))^2/h/h;
fr = y_1(index)*(t-x(index))*(t-x(index+1))^2/h/h + ...
y_1(index+1)*(t-x(index+1))*(t-x(index))^2/h/h;
f = fl + fr; %x0所在区间的插值函数
f0 = subs(f,'t',x0); %x0处的插值
function [f,f0] = ThrSample1 (x,y,y_1, y_N,x0)
syms t;
f = 0.0;
f0 = 0.0;
if(length(x) == length(y))
n = length(x);
else
disp('x和y的维数不相等！');
return;
end %维数检查
for i=1:n
if(x(i)<=x0)&& (x(i+1)>=x0)
index = i;
break;
end
end %找到x0所在区间
for i=1:n
if(x(i)<=x0)&& (x(i+1)>=x0)
index = i;
break;
end
end %找到x0所在区间
A = diag(2*ones(1,n)); %求解m的系数矩阵
u = zeros(n-2,1);
lamda = zeros(n-1,1);
c = zeros(n,1);
for i=2:n-1
u(i-1) = (x(i)-x(i-1))/(x(i+1)-x(i-1));
lamda(i) = (x(i+1)-x(i))/(x(i+1)-x(i-1));
c(i) = 3*lamda(i)*(y(i)-y(i-1))/(x(i)-x(i-1))+ ...
3*u(i-1)*(y(i+1)-y(i))/(x(i+1)-x(i));
A(i, i+1) = u(i-1);
A(i, i-1) = lamda(i); %形成系数矩阵及向量c
end
c(1) = 2*y_1;
c(n) = 2*y_N;
m = followup(A,c); %用追赶法求解方程组
h = x(index+1) - x(index); %x0所在区间长度
f = y(index)*(2*(t-x(index))+h)*(t-x(index+1))^2/h/h/h + ...
y(index+1)*(2*(x(index+1)-t)+h)*(t-x(index))^2/h/h/h + ...
m(index)*(t-x(index))*(x(index+1)-t)^2/h/h - ...
m(index+1)*(x(index+1)-t)*(t-x(index))^2/h/h;
%x0所在区间的插值函数
f0 = subs(f,'t',x0); %x0处的插值
function [f,f0] = ThrSample2 (x,y,y2_1, y2_N,x0)
syms t;
f = 0.0;
f0 = 0.0;
if(length(x) == length(y))
n = length(x);
else
disp('x和y的维数不相等！');
return;
end %维数检查
for i=1:n
if(x(i)<=x0)&& (x(i+1)>=x0)
index = i;
break;
end
end %找到x0所在区间
for i=1:n
if(x(i)<=x0)&& (x(i+1)>=x0)
index = i;
break;
end
end %找到x0所在区间
A = diag(2*ones(1,n)); %求解m的系数矩阵
A(1,2) = 1;
A(n,n-1) = 1;
u = zeros(n-2,1);
lamda = zeros(n-1,1);
c = zeros(n,1);
for i=2:n-1
u(i-1) = (x(i)-x(i-1))/(x(i+1)-x(i-1));
lamda(i) = (x(i+1)-x(i))/(x(i+1)-x(i-1));
c(i) = 3*lamda(i)*(y(i)-y(i-1))/(x(i)-x(i-1))+ ...
3*u(i-1)*(y(i+1)-y(i))/(x(i+1)-x(i));
A(i, i+1) = u(i-1);
A(i, i-1) = lamda(i); %形成系数矩阵及向量c
end
c(1) = 3*(y(2)-y(1))/(x(2)-x(1))-(x(2)-x(1))*y2_1/2;
c(n) = 3*(y(n)-y(n-1))/(x(n)-x(n-1))-(x(n)-x(n-1))*y2_N/2;
m = followup(A,c); %用追赶法求解方程组
h = x(index+1) - x(index); %x0所在区间长度
f = y(index)*(2*(t-x(index))+h)*(t-x(index+1))^2/h/h/h + ...
y(index+1)*(2*(x(index+1)-t)+h)*(t-x(index))^2/h/h/h + ...
m(index)*(t-x(index))*(x(index+1)-t)^2/h/h - ...
m(index+1)*(x(index+1)-t)*(t-x(index))^2/h/h;
%x0所在区间的插值函数
f0 = subs(f,'t',x0); %x0处的插值
function [f,f0] = ThrSample3 (x,y,x0)
syms t;
f = 0.0;
f0 = 0.0;
if(length(x) == length(y))
n = length(x);
else
disp('x和y的维数不相等！');
return;
end %维数检查
for i=1:n
if(x(i)<=x0)&& (x(i+1)>=x0)
index = i;
break;
end
end %找到x0所在区间
for i=1:n
if(x(i)<=x0)&& (x(i+1)>=x0)
index = i;
break;
end
end %找到x0所在区间
A = diag(2*ones(1,n-1)); %求解m的系数矩阵
h0 = x(2)-x(1);
h1 = x(3)–x(2);
he = x(n)-x(n-1);
A(1,2) = h0/(h0+h1);
A(1,n-1) = 1 - A(1,2);
A(n-1,1) = he/(h0+he);
A(n-1,n-2) = 1 - A(n-1,1);
c = zeros(n-1,1);
c(1) = 3* A(1,n-1)*(y(2)-y(1))/h0 + 3* A(1,2)*(y(3)-y(2))/h1;
for i=2:n-2
u = (x(i)-x(i-1))/(x(i+1)-x(i-1));
lamda = (x(i+1)-x(i))/(x(i+1)-x(i-1));
c(i) = 3*lamda*(y(i)-y(i-1))/(x(i)-x(i-1))+ ...
3*u*(y(i+1)-y(i))/(x(i+1)-x(i));
A(i, i+1) = u;
A(i, i-1) = lamda; %形成系数矩阵及向量c
end
c(n-1) = 3*( he*(y(2)-y(1))/h0+h0*( y(n)-y(n-1))/he)/(h0+he);
m = zeros(n,1);
[m(2:n),Q,R] = qrxq(A,c); %用qr分解法法求解方程组
m(1) = m(n);
h = x(index+1) - x(index); %x0所在区间长度
f = y(index)*(2*(t-x(index))+h)*(t-x(index+1))^2/h/h/h + ...
y(index+1)*(2*(x(index+1)-t)+h)*(t-x(index))^2/h/h/h + ...
m(index)*(t-x(index))*(x(index+1)-t)^2/h/h - ...
m(index+1)*(x(index+1)-t)*(t-x(index))^2/h/h;
%x0所在区间的插值函数
f0 = subs(f,'t',x0); %x0处的插值